On the weight spaces of Lie algebra modules and their Jordan kernel (Q579387)

Let L be a finite-dimensional Lie algebra over a field k of characteristic zero and let V be an L module. Put \(V_{\lambda}=\{v\in V|\forall x\in L\), \(xv=\lambda (x)v\}\) and \(V^{\lambda}=\{v\in V|\forall x\in L\), \(\exists n\in {\mathbb{N}}\), \((x-\lambda (x))^ nv=0\}\) where \(\lambda\) is a function from L to k. In the case V is finite-dimensional, \textit{M. K. Smith} [Proc. Am. Math. Soc. 95, 524-526 (1985; Zbl 0603.17004)] has proved that \(V_{\lambda}\) must be nonzero if \(V^{\lambda}\) is nonzero and \(V^{\lambda}\) is a submodule of V. In this paper, these results have been generalized to arbitrary L-modules V and a new characterization of these weight spaces \(V^{\lambda}\) which simplifies their actual computation has been given. Particularly, the details for the case V is either the universal enveloping algebra U(L) of L or its division ring of quotients D(L) have been discussed.